1. Field of the Invention
The present invention relates to a Fizeau interferometer and a measurement method using the Fizeau interferometer.
2. Description of the Related Art
Conventionally, a Fizeau interferometer is known, which is equipped with a reference surface for reflecting part of the laser light emitted from a laser light source as reference light, for allowing part of the other part of the laser light to pass through as measurement light and for emitting the measurement light to a surface to be measured, and a measuring apparatus for measuring the form of the surface to be measured on the basis of the interferograms of the reference light and the measurement light reflected by the surface to be measured (for example, refer to JP-A-2007-333428).
The Fizeau interferometer described in JP-A-2007-333428 is equipped with a reference surface and a computer (measuring apparatus) and measures the form of a surface to be measured using a phase shift method. The phase shift method is a method in which the position of a reference surface is moved along the optical axis of laser light to obtain the intensities of interferograms at a plurality of positions and to measure the form of the surface to be measured on the basis of the intensities of these interferograms. Furthermore, this kind of Fizeau interferometer can measure a spherical surface to be measured by forming the reference surface into a spherical surface. In the following descriptions, a reference surface being spherical is referred to as a reference spherical surface, and a surface to be measured having a spherical form is referred to as a spherical surface to be measured.
FIG. 8 is a schematic view showing a state in which the form of the surface of a spherical body is measured using the Fizeau interferometer. More specifically, FIG. 8 is a schematic sectional view sectioned along an xy plane wherein the optical axis direction of the laser light is an x-axis direction and an axis orthogonal to this x-axis is a y-axis. This is because it can be assumed that the ideal measurement optical system of the Fizeau interferometer is rotationally symmetric around the optical axis (x-axis) and is represented in a two-dimensional plane. As shown in FIG. 8, a Fizeau interferometer 100 is equipped with an optical element 110 having a reference spherical surface 111 with a radius R and a measuring apparatus (not shown) and is used to measure the form of the surface of a spherical body 120 with a radius r (hereafter referred to as a spherical surface 121 to be measured). In this figure, a laser light source (not shown) is disposed in the plus x-axis direction (on the right side in FIG. 8) of the optical element 110 and emits laser light in the minus x-axis direction. In FIG. 8, the laser light source emits laser light in the range enclosed by solid lines L1 and L2.
Part of the laser light emitted from the laser light source is reflected by the reference spherical surface 111 and serves as reference light, and part of the other part of the laser light passes through the reference spherical surface 111 and serves as a measurement light. The measurement light is reflected by the spherical surface 121 to be measured, and part of the measurement light reflected by the spherical surface 121 to be measured passes through the reference spherical surface 111. An interferogram is generated by the reference light reflected by the reference spherical surface 111 and the measurement light reflected by the spherical surface 121 to be measured. The measuring apparatus measures the form of the spherical surface 121 to be measured on the basis of the intensity of this interferogram.
Since the Fizeau interferometer 100 measures the displacement between the reference spherical surface 111 and the spherical surface 121 to be measured on the basis of the intensity of the interferogram, the focal point of the reference spherical surface 111 is required to be aligned with the focal point of the spherical surface 121 to be measured, i.e., the center of the spherical body 120. For this reason, when the Fizeau interferometer 100 is used to measure the form of the spherical surface 121 to be measured, an adjustment is carried out in advance so that the focal point of the reference spherical surface 111 is aligned with the center of the spherical body 120. In FIG. 8, the focal point of the reference spherical surface 111 and the center of the spherical body 120 are placed at the origin O of the orthogonal coordinate system shown in FIG. 8.
Hence, for example, in the laser light emitted from the laser light source and entering the optical element 110, the laser light ray passing through point P1 on the reference spherical surface 111, i.e., the measurement light, is emitted in the direction to the origin O. The measurement light is reflected at point P2 on the spherical surface 121 to be measured and enters the optical element 110 at the point P1 on the reference spherical surface 111. Furthermore, interferogram is generated by the reference light reflected at the point P1 and the measurement light entering the optical element 110 at the point P1 and is emitted from the optical element 110 in the plus x-axis direction.
In other words, the measurement light emitted from the reference spherical surface 111 reciprocally travels through the optical paths between the reference spherical surface 111 and the spherical surface 121 to be measured. The measuring apparatus uses a CCD (charge-coupled device) camera (not shown) to image the interferogram of the reference light emitted from the optical element 110 and the measurement light and to obtain an interference fringe image based on optical path difference (OPD), i.e., the intensity of the interferogram. The measuring apparatus then observes this interference fringe image, thereby measuring the form of the spherical surface 121 to be measured. When it is herein assumed that the angle of the point P1 (hereafter referred to as an observation angle) is θ, the coordinates of the point P1 are represented by (R cos θ, R sin θ), and the coordinates of the point P2 are represented by (r cos θ, r sin θ). Hence, the optical path difference OPD (θ) between the reference light and the measurement light can be represented by the following expression (1).OPD(θ)=2√{square root over ((R cos θ−r cos θ)2+(R sin θ−r sin θ)2)}{square root over ((R cos θ−r cos θ)2+(R sin θ−r sin θ)2)}  (1)
FIG. 9 is a schematic view showing a state in which the position of the reference spherical surface 111 is moved so that the form of the spherical surface 121 to be measured is measured using the phase shift method. When the position of the reference spherical surface 111 is moved along the optical axis of the laser light so as to become close to the spherical body 120 by a distance δ as shown in FIG. 9, the focal point F of the reference spherical surface 111 is moved y the distance δ from the origin O b in the minus x-axis direction. Hence, the coordinates of the point P1 are represented by (R cos θ−δ, R sin θ). Furthermore, since the measurement light passing through the point P1 is emitted in the direction to the focal point F, the measurement light is reflected at point P2′ on the spherical surface 121 to be measured. When it is herein assumed that the angle of the point P2′ is θ′, the coordinates of the point P2′ are represented by (r cos θ′, r sin θ′). Hence, the optical path difference OPD (θθ′) between the reference light and the measurement light can be represented by the following expression (2).OPD(θθ′)shift=2√{square root over ((R cos θ−δ−r cos θ′)2+(R sin θ−r sin θ′)2)}{square root over ((R cos θ−δ−r cos θ′)2+(R sin θ−r sin θ′)2)}  (2)
In the phase shift method, the distance δ (hereafter referred to as a movement amount δ) through which the position of the reference spherical surface 111 is moved is generally assumed to be approximately half of the wavelength of the laser light emitted from the laser light source or at most approximately two times of the wavelength. In the case that the movement amount δ is set in this way, the change in the observation angle θ due to the displacement of the point P2 at which the measurement light is reflected to the point P2′ is very small in comparison with the distance between the pixels in an ordinary CCD camera and rarely causes significant change in the interference fringe image. In other words, it can be assumed that θ is nearly equal to θ′, and the above-mentioned expression (2) can be replaced with the following expression (3).OPD(θ)shift=2√{square root over ((R cos θ−δ−r cos θ)2+(R sin θ−r sin θ)2)}{square root over ((R cos θ−δ−r cos θ)2+(R sin θ−r sin θ)2)}  (3)
FIG. 10 is a graph showing the relationship between the amount of change in the optical path difference OPD and the observation angle θ at the time when the movement amount δ is changed at constant intervals. In FIG. 10, the vertical axis represents the amount of change in the optical path difference OPD and the horizontal axis represents the observation angle θ. Furthermore, FIG. 10 shows graphs G71 to G79 at the time when the wavelength of the laser light is λ(=633 nm) and when the movement amount δ is changed from 0 to λ at λ/8 intervals. FIG. 11 is a graph showing the relationship between the optical path difference OPD and the movement amount δ at the time when the observation angle θ is changed at constant intervals. In FIG. 11, the vertical axis represents the optical path difference OPD and the horizontal axis represents the movement amount δ. Furthermore, FIG. 11 shows graphs G81 to G84 at the time when the adjustment is carried out in advance so that the focal point of the reference spherical surface 111 is aligned with the center of the spherical body 120, while the optical path difference OPD is 80 mm and the observation angle θ is changed 0°, 20°, 40° and 50°. In FIGS. 10 and 11, the radius R of the reference spherical surface 111 is 50 mm and the radius r of the spherical body 120 is 10 mm.
For example, in the graph G79 wherein δ is equal to λ, the amount of change in the optical path difference OPD is approximately 2λ when the observation angle θ is 0° as shown in FIG. 10. In other words, when the observation angle θ is 0°, the amount of change in the optical path difference OPD is approximately two times of the movement amount δ, an expected amount of change. This is based on the fact that the measurement light emitted from the reference spherical surface 111 reciprocally travels between the reference spherical surface 111 and the spherical surface 121 to be measured. In addition, as shown in FIG. 11, when it is assumed that the observation angle θ is constant, the optical path difference OPD changes nearly linearly. More specifically, the optical path difference OPD becomes smaller as the position of the reference spherical surface 111 is moved so as to become closer to the spherical body 120 (the movement amount δ is in the plus direction) and becomes larger as the position of the reference spherical surface 111 is moved so as to become away from the spherical body 120 (the movement amount δ is in the minus direction).
On the other hand, as the observation angle θ becomes larger, the amount of change in the optical path difference OPD becomes smaller than 2λ as shown in FIG. 10. For example, the amount of change in the optical path difference OPD at the time when the observation angle θ is 50° is approximately ⅔ of the amount of change at the time when the observation angle θ is 0°. In other words, the amount of change in the optical path difference OPD differs depending on the observation angle θ even in the case that the movement amount δ is the same. Generally speaking, in the phase shift method, the form of the surface to be measured is calculated by applying a specific algorithm to the amount of change in the intensity of the interferogram corresponding to the amount of change in the phase of an interference fringe. Furthermore, since the phase of the interference fringe changes depending on the change in the optical path difference OPD, in the case that the amount of change in the optical path difference OPD does not become its expected amount of change, the amount of change in the phase of the interference fringe does not become its expected amount of change. As a result, an error corresponding to the observation angle θ occurs when the form of the spherical surface 121 to be measured is measured using the phase shift method.
Next, the relationship between a measurement error occurring in the phase shift method and the observation angle θ will be examined. The relationship between the intensity I of the interferogram and the movement amount δ can be represented by the following expression (4).
                              I          i                =                              I            0                    +                      A            ⁢                                                  ⁢                          cos              ⁡                              (                                                                                                    OPD                        ⁡                                                  (                          θ                          )                                                                                            δ                        ⁢                                                                                                  ⁢                        i                                                              ·                                                                  2                        ⁢                        π                                            λ                                                        +                                      ϕ                    ⁡                                          (                      θ                      )                                                                      )                                                                        (        4        )            
In this expression, φ(θ) is the initial phase angle of the fluctuation component of a signal based on the intensity of the interferogram and is the difference between the reference spherical surface 111 and the spherical surface 121 to be measured when the adjustment is carried out in advance so that the focal point of the reference spherical surface 111 is aligned with the center of the spherical body 120, i.e., the form of the spherical surface 121 to be measured. Furthermore, Io is an offset of a signal based on the intensity of the interferogram, and A is the amplitude of the fluctuation component of this signal. Moreover, the first term of the cos function represents a known value that is obtained by converting the optical path difference OPD(θ) at the time when the reference spherical surface 111 is located at the position (movement amount δi) indicated by a suffix i into a phase based on the wavelength λ of the laser light. Hence, when it is assumed that the intensity of the interferogram at the time when the reference spherical surface 111 is located at the position indicated by a suffix i is Ii, since unknown quantities in the expression (4) are three quantities, i.e., Io, A and φ(θ), once the intensities Ii of at least three interferograms are obtained, φ(θ), i.e., the form of the spherical surface to be measured, can be known by solving simultaneous equations.
Furthermore, in the phase shift method, numerous methods have been proposed with respect to the combinations of the movement amounts δi for obtaining φ(θ), i.e., the form of the spherical surface to be measured, and specific algorithms corresponding thereto. For example, φ(θ) can be calculated by using such algorithms as represented in the following expressions (5) and (6) (Daniel Malacara, Optical Shop Testing Third Edition, Wiley Interscience, 2007, pp 594-595).
                              ϕ          ⁡                      (            θ            )                          =                              tan                          -              1                                ⁡                      [                                          2                ⁢                                  (                                                            I                      4                                        -                                          I                      2                                                        )                                                                              I                  1                                -                                  2                  ⁢                                      I                    3                                                  +                                  I                  5                                                      ]                                              (        5        )                                          ϕ          ⁡                      (            θ            )                          =                              tan                          -              1                                ⁡                      [                                                            7                  ⁢                                      (                                                                  I                        3                                            -                                              I                        5                                                              )                                                  -                                  (                                                            I                      1                                        -                                          I                      7                                                        )                                                                              8                  ⁢                                      I                    4                                                  -                                  4                  ⁢                                      (                                                                  I                        2                                            +                                              I                        6                                                              )                                                                        ]                                              (        6        )            
More specifically, the expression (5) represents an algorithm (hereafter referred to as a 5-step method) that is used when the intensities of the interferograms at five positions are obtained by changing the movement amount δ from 0 at λ/8 intervals so that the phase of the interference fringe is changed at π/2 intervals. The expression (6) represents an algorithm (hereafter referred to as a 7-step method) that is used when the intensities of the interferograms at seven positions are obtained by changing the movement amount δ from 0 at λ/8 intervals so that the phase of the interference fringe is changed at n/2 intervals.
FIG. 12 is a graph showing the relationship between a measurement error occurring in the phase shift method and the observation angle θ. In FIG. 12, it is assumed that the spherical surface 121 to be measured is an ideal spherical surface, and the measurement error occurring in the 5-step method is shown in a graph G91 and the measurement error occurring in the 7-step method is shown in a graph G92. Furthermore, in FIG. 12, the vertical axis represents the measurement error, and the horizontal axis represents the observation angle θ.
As described above, when the observation angle θ is 0°, the amount of change in the optical path difference OFD has become its expected amount of change. Hence, the measurement error is 0 as shown in FIG. 12.
On the other hand, when the observation angle θ is 50° for example, the amount of change in the optical path difference OPD is approximately ⅔ of the amount of change at the time when the observation angle θ is 0° as described above. Hence, the error in form is larger than that at the time when the observation angle θ is 0°. In addition, the measurement error occurring in the 7-step method is larger than the measurement error occurring in the 5-step method.
As described above, in the case that the phase shift method is used when the form of the surface of the spherical body 120 is measured using the Fizeau interferometer 100, the amount of change in the optical path difference OPD does not become its expected value due to the relationship to the observation angle θ but generates a measurement error, whereby there is a problem that the form of the spherical surface 121 to be measured cannot be measured properly.